[docs-infra] Migrate docs to Next App Router

[oliviertassinari]

Summary

This issue is about moving our docs away from the legacy Pages Router of Next.js to App Router.

Motivation

1. It's critical as most new projects created by developers today are on the App Router,

https://analytics.google.com/analytics/web/#/analysis/p353089763/edit/IBVRQAiCT2mrgZwzpg9fFQ

How can we build a great DX for something we don't experience every day? It's impossible IMHO.

2. It's a great way to solve the Layout piece of [docs-infra] Fix page change speed regression #41117.

Context

It's related to #38966.

Search keywords: App Router

[flaviendelangle]

@samuelsycamore posted this article https://www.flightcontrol.dev/blog/nextjs-app-router-migration-the-good-bad-and-ugly on Slack

Ugly: abysmal dev server performance

We need to see how bad it is 😬

[oliviertassinari]

@flaviendelangle Thanks for adding this here. 👍 to measure the performance in dev mode as we do our migration. I hope zero runtime CSS-in-JS won't add to the problem 😄.

Now this is part of the value, e.g. #39432, to experience that first hand, and fix what we can on our end.